






































  
Analytical Approximations for the Principal Branch of the Lambert W Function

Roy M. Howard

School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, GPO 
Box U1987, Perth, 6845, Australia

ABSTRACT. A geometric based approach for specifying approximations to the Lambert W function,
which can achieve any set relative error bound over the interval , is detailed. Approximations
that can achieve arbitrarily high accuracy for the interval , based on a two point spline
approximation, are specified. Iterative methods can be used to improve the accuracy of the
approximations. 

Applications include, first, analytical expressions, with set relative error bounds, for the Lambert W
function over the interval . Second, approximations, with an arbitrarily low relative error, for
upper and lower bounds for the Lambert W function. Third, analytical expressions for the evaluation
of  and the integral of , for , without knowledge of . Fourth, a direct
approach for evaluating the Lambert W function to achieve a prior set error constraint. 

1.  INTRODUCTION

The Lambert W function is associated with Lambert [19] and is a multivalued complex
function with the single valued function, associated with the  branch, being denoted

. The principle branch of the Lambert W function, , is the function defined by the

inverse of  for the case of , i.e.

(1)

The Lambert W function does not have an explicit analytical form but is of importance
consistent with increasing applications as detailed in the literature, e.g. [7], [3], [25], [9],
[6], [14], [26], [20], [18], [4], and [11]. Generalizations of the Lambert W function are also of
interest, e.g. [8] and [22]. Efficient numerical methods for computing values of the Lambert
W function have long been known, e.g. [10], and with an advance detailed by Fukushima
[12].

The focus of this paper is on the principle branch and the real case which continues
to receive research interest, e.g. [17]. The graph of the Lambert W function, for this case,
is shown in Figure 1 and, for notational simplicity, is denoted  in this paper. Existing
analytical approximations for the principle branch and real case of the Lambert W func-
tion, e.g. [5], [3] and [17], are, in general, custom and cannot be directly generalized to
obtain approximations of arbitrarily high accuracy. This paper provides a geometrical
basis for defining such approximations.  

0  
1 e– 0 

0 

W y  W y  y 0   W y 

kth

Wk W0

y f x  xe
x

= = Re x  1–

x W0 y  f
1–
y .= =

W

Correspondence: r.howard@curtin.edu.au 

https://adac.ee/
https://doi.org/10.28924/ada/ma.2.14


 

  

The advances in this paper are twofold. First, a systematic geometric approach for
defining approximations to the Lambert W function, over the interval , with quad-
ratic convergence. Convergence is proved. The approximations can be used to specify,
with an arbitrarily low relative error, upper and lower bounds for the Lambert W function.
Second, a systematic method for combining series expansions at , and the origin, to
define arbitrarily accurate approximations for the interval . Applications of the
approximations include analytical expressions, with set relative error bounds, for the
Lambert W function over the interval , evaluation of  and the integral of

, for , without knowledge of , and a direct approach for evaluating
the Lambert W function to achieve a prior defined error. 

A review of published approximations for the Lambert W function is provided in sec-
tion 2. The proposed geometric approach for establishing approximations to the Lambert
W function is detailed in section 3. Convergence is discussed in section 4 and in section
5 convergent two point spline based approximations, for the interval , are
detailed. The use of iterative methods, to improve the accuracy of approximations, is dis-
cussed in Section 6. Applications are detailed in section 7 and conclusions are stated in
section 8.

1.1.  Notation and Properties. The notation of  is used which is consistent with
the principle value of the Lambert W being the inverse function of , ,

. Relevant properties of the Lambert W function are detailed in Appendix A.
For an arbitrary function , defined over the interval , an approximating function

 has a relative error, at a point , defined according to . The rel-
ative error bound for the approximating function, over the interval , is defined
according to

(2)

The notation  is used.

Mathematica has been used to facilitate analysis and to obtain numerical results. In
general, relative error results, associated with approximations to the Lambert W function,
have been obtained by sampling specified intervals, in either a linear or a logarithmic
manner, as appropriate, with  points.

FIGURE 1.  Graph of  and its inverse which is the
Lambert W function for the principle branch and the real case.

f x  xe
x

=

W y 

x y

f x  xe
x

=

1– 1 e– 

1 e– 1– 

0  

1 e–

1 e– 0 

0  W y 
W y  y 0   W y 

1 e– 0 

x W y =

y f x  xe
x

= = x 1–
y 1 e–

f   
fA x1 re x1  1 fA x1  f x1 –=

  

reB max re x1  : x1    .=

f
k 

x 
x
k

k

d

d f x =

1000

https://doi.org/10.28924/ada/ma.2.14


 

  2.  PUBLISHED APPROXIMATIONS

A Taylor series expansion, at the origin, for the Lambert W function is well known, e.g.
[17], eqn. 2, and yields the following approximation which has a limited region of conver-
gence:

(3)

The relationship  implies  (positive sign for ; negative
sign for ) and, hence:

(4)

Such approximations suggest the more general approximation for the Lambert W function
of

(5)

Fritsch et al., [10], eqn. 6, utilizes an initial approximation of  which is suitable
for .

2.1.  Published Approximations. The following is an overview of indicative published
approximations for the Lambert W function. Additional useful references include [26] and
[14].

Boyd [5], eqn. 5-7, proposed the approximation

(6)

which is valid for . Whilst the approximation is sharp at , it is not zero at
the origin and, thus, is not sharp at this point. It has a relative error bound for the inter-
val  of .

Barry et al. [3], eqn. 12, proposed the approximation

(7)

for . This approximation was modified, [3], eqn. 15, to be valid for  according
to

(8)

T y  y y
2

– 3y
3

2
-------- 8y

4

3
--------– 125y

5

24
-------------- 54y

6

5
----------- 16807y

7

720
--------------------+– 16384y

8

315
--------------------– + + += y 1

e
---.

y xe
x

= x x ln+ y ln= x y 0
x y 0

x
y y 1«

y ln y 1.»





x W y  1 y+ ln = y 1–
e

------.

W y  y ln
y e

WBd y  1 2 1 ey+ 
10 ln 10 ln ln–

------------------------------------------------- 11 ey+ ln 11 ey+ ln ln–tanh+– =

1 1
10
------ 1 ey+ ln 7

5
---– 3–

40
------ 1 ey+ ln 7

5
---–

2
exp+

y 1 e– y 1 e–=

1   0.0499

WB1
y  6

5
--- y

12
5

------ y
1 12y 5+ ln

-----------------------------------ln

---------------------------------------------------------ln=

y 0 y 1 e–

WB2
y  1 +  6

5
--- y

12
5

------ y
1 12y 5+ ln

-----------------------------------ln

---------------------------------------------------------ln  2y
1 2y+ ln

-------------------------ln –=  0.4586887.=

https://doi.org/10.28924/ada/ma.2.14


 

  
The relative error bound, for , is .

Iacono and Boyd, [17], eqn. 17, proposed the following approximation which is valid
for 

(9)

This approximation yields a relative error bound of  for . An improved
approximation, [17], eqn. 19, 20, is

(10)

which yields a relative error bound for  of  (the bound occurs at  of the
order of ) for the case of  optimally chosen as . The approximation is sharp
at .

2.1.1.  Padè Approximations. Padè approximates for the Lambert W function for the
interval  have been proposed, e.g. [21], eqn. 34:

(11)

Higher order Padè approximates are detailed in Fukushima [13].

2.1.2.  Comparison of Approximations. The relative errors in the above specified
approximations are shown in Figure 2 and Figure 3.            

2.2.  Classic Iterative Approximations. The classical iterative approximation for the
Lambert W function, e.g. [17], eqn. 8-10, is based on the fundamental relationship

, , and an initial approximation  as specified in

y 0 1.96 10
3–

y 1– e  

WI1
y  1 y

1 1 y+ ln
2

----------------------+
--------------------------------+ .ln=

3.53 10
2– y 0  

WI2
y  1– a 1 b 1 ey++

1 c 1 1 ey++ ln+
---------------------------------------------------ln +=

c e
1 a

1– 2 a–

1 2 e1 a
ln–

----------------------------------------= b 2
a

------- c+ =

y 0 4.53 10
3– y

10
12

a a 2.036=

y 1 e–=

y 1– e 1 

WL y 
1 123y

40
------------ 21y

2

10
-----------+ +

1 143y
40

------------ 713y
2

240
--------------+ +

------------------------------------------ 1 y+ .ln=

FIGURE 2.  Graph of the relative errors, over the interval
, in published approximations to .1– e 1  W y 

y

re y 

WI1
y 

WI2
y 

WB2
y 

WBd y 

WL y 

WL y 

WB2
y 

WI1
y 

x y ln x ln–= x y 0 x W0 y  1 y+ ln=

https://doi.org/10.28924/ada/ma.2.14


 

  
Equation 5. A first order approximation arises by substitution of this approximation into
the expression  to yield   

(12)

Iteration yields

(13)

In general:

(14)

2.2.1.  Comparison of Approximations. The relative error in the iterative approximations,
of orders zero to three, are detailed in Figure 4 and Figure 5. The relative errors
decrease, for large values of , at an increasing rate as the order of approximation is
increased. The approximations are poor for  which is consistent with the
assumptions made in the iteration.       

2.2.2.  Alternative Iterative Approximations. An alternative iterative approach is to utilize
the relationship  and solve for the error given an initial approximation

FIGURE 3.  Graph of the relative errors in published
approximations to .W y 

y

re y 

WI1
y 

WI2
y 

WBd y  WL y 

WB2
y 

WL y 

x ln

W1 y  y ln 1 y+ ln ln– y
1 y+ ln

---------------------- .ln= =

W2 y  y ln y ln 1 y+ ln ln–ln– y

y
1 y+ ln

----------------------ln

---------------------------------ln .= =

Wi y  y ln Wi 1– y  ln– = i 1 2 3     W0 y  1 y+ .ln=

y

y 10

FIGURE 4.   Graph of the relative errors, over the interval
, in iterative approximations to . The relative errors

in the approximations for  and  are high.
1– e 1  W y 

W2 W3

y

WI3
y 

re y 

W0 y 

W1 y 
W2 y 

x y ln x ln–=

https://doi.org/10.28924/ada/ma.2.14


 

  
of . For  fixed, consider an initial approximation of , with an error of , which
implies  and, thus, e.g. [17], eqn. 11:

(15)

For the case where the error is small and , a first order Taylor series for the
logarithm function yields 

(16)

and the first order approximation

(17)

The general iteration formula, e.g. [17], eqn. 12 

(18)

then follows. Higher order iteration, based on a higher order approximation for
, is detailed in [10].

With a starting value, utilized in Fritsch et al. [10], of  (suitable for ) it
follows that a first order approximation is

(19)

The relative error in this approximation is shown in Figure 5 and the relative error bound
for the interval  is . 

A first order iteration, based on Equation 18, with , yields

the approximation [17], eqn. 18:

FIGURE 5.  Graph of the relative errors in iterative approximations
to .W y 

y

W0 y 

W2 y 

W3 y 

W1 y 

WI3
y 

re y 

WF y  WF y 

x0 y x0 0

y x0 0+  x0 0+ exp=

x0 0+ y ln x0 0+  ln– y
x0
-----ln 1

0

x0
-----+ .ln–= =

0 x0 1«

x0 0+ y
x0
-----ln

0

x0
-----–  0

x0

1 x0+
-------------- y

x0
-----ln x0– 

x1 x0 0+
x0

1 x0+
-------------- 1 y

x0
-----ln+ .=

xi 1+

xi
1 xi+
------------- 1 y

xi
----ln+ =

1 0 x0+ ln

x0 y ln= y e

WF y  y ln
1 y ln+
---------------------- 1 y

y ln
-------------ln+ .=

e   1.47 10
2–

x0 y  1 y
1 0.5 1 y+ ln+
---------------------------------------+ln=

https://doi.org/10.28924/ada/ma.2.14


 

  
(20)

The relative error in this approximation is shown in Figure 4 and Figure 5. The relative
error bound, over the interval , is .

2.3.  Approximations via Newton-Raphson Iteration. Consistent with the illustration
shown in Figure 6, a direct Newton-Raphson method for solving for , in the equation

, is:

(21)

A first iteration, based on a known approximating function  for , is, e.g. [17], eqn. 15:

(22)

Halley’s method can similarly be utilized, e.g. [26].    
With an initial value of , the first and second order approximations,

respectively, are:

(23)

(24)

2.3.1.  Results. Graphs of the variation of the relative error, with iteration level, are
shown in Figure 7 and Figure 8 for the case of an initial approximation of .

WI3
y 

1 y
1 0.5 1 y+ ln+
---------------------------------------+ln

1 1 y
1 0.5 1 y+ ln+
---------------------------------------+ln+

--------------------------------------------------------------------- 1 y

1 y
1 0.5 1 y+ ln+
---------------------------------------+ln

------------------------------------------------------------ln+ .=

0   2.16 10
4–

x

y xe
x

=

xi xi 1–

xi 1– e
xi 1– y–

e
xi 1– xi 1– e

xi 1–+
-----------------------------------------– xi 1–

xi 1– ye
xi 1––

–

1 xi 1–+
----------------------------------.–= =

g W

W y  g y  g y  g y  exp y–
g y  exp g y  g y  exp+

--------------------------------------------------------------------– g y  g y  y g– y  exp–
1 g y +

------------------------------------------------.–=

FIGURE 6.  Newton-Raphson iteration for determining an
approximation to the solution, denoted , of  for 

fixed and based on an initial value of .

xo y x x exp= y

x0

x
xo

h x0 

x0x1

h x  xe
x

y–=

x2

h x1 

x0 1 y+ ln=

WN1
y  1 y+ ln 1 y+ ln y 1 y+ –

1 1 y+ ln+
----------------------------------------------------– =

WN2
y  1 y+ ln 1 y+ ln y 1 y+ –

1 1 y+ ln+
----------------------------------------------------– –=

1 y+ ln 1 y+ ln y 1 y+ –
1 1 y+ ln+

----------------------------------------------------– y
1 y+
------------ 1 y+ ln y 1 y+ –

1 1 y+ ln+
----------------------------------------------------exp–

1 1 y+ ln 1 y+ ln y 1 y+ –
1 1 y+ ln+

----------------------------------------------------–+
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------- .

x0 1 y+ ln=

https://doi.org/10.28924/ada/ma.2.14


 

  
Note, for higher order iteration, the relative error increases, from an increasingly low
level, as  increases.          

3.  GEOMETRIC BASIS FOR ITERATIVE APPROXIMATIONS TO LAMBERT W 
FUNCTION

3.1.  Geometric Basis. To establish a systematic, geometrically based, approach for
establishing approximations to the Lambert W function of arbitrarily high accuracy,
consider a set value of . The Lambert W function associated with , denoted , is such

that  and is the point defined by the intersection of the two curves  and 
as illustrated in Figure 9 for the case of . The geometry associated with this
intersection of the two curves is the basis for an initial approximation and for the
iterative approximations detailed in the following theorem. 

Theorem 3.1.   Iterative Approximations for Lambert W Function. For  fixed, ,
approximations to the Lambert W function can be iteratively defined according to

(25)

y

FIGURE 7.   Graph of the relative errors, in the zero to fourth order
iterative approximations to , based on Newton-Raphson iteration

with an initial approximation of .

W y 

WN0
y  1 y+ ln=

y

WN1
y 

re y 

WN2
y 

WN4
y 

WN3
y 

WN0
y 

FIGURE 8.   Graph of the relative errors, in the zero to fifth order
iterative approximations to , based on Newton-Raphson iteration

with an initial approximation of .

W y 

WN0
y  1 y+ ln=

y

WN0
y 

re y 

WN3
y 

WN4
y 

WN5
y 

y y xo

xo ye
xo–

= x ye
x–

y 0

y y 1 e–

WLi

WLi 1–
1 WUi 1–

+ 

1 WLi 1–
+

--------------------------------------------= i 1 2    WL0
y=

WUi

y
WLi

---------ln = i 1 2    WU0
0.=

https://doi.org/10.28924/ada/ma.2.14


 

  Proof. Consider the geometry detailed in Figure 9, for the case of , and an initial
approximation to  of  which is based the intersection of the first order

Taylor series for  at the origin, i.e. , and . An associated approximation,
denoted , arises from the intersection of the level defined by  and the curve .
The solution is   

   (26)

Second order approximations follow from the intersection of  with a first order Taylor
series of , based on the point , i.e. , to yield

(27)

and by the intersection of this level with the curve  to yield

(28)

The general iterative form:

(29)

then follows. Whilst the geometry, and the defined approximations, are clearly defined for
the case of , the approximations are also valid for  as simulation results,
shown in Figure 10, demonstrate.

3.1.1.  Explicit Approximations. Approximations to the Lambert W function, of orders one
to five, are:

(30)

y 0
xo WL1

y 1 y+ =

ye
x–

y yx– x

WU1
WL1

ye
x–

FIGURE 9.   Illustration of the geometry underpinning an iterative relationship to find
an approximation to  which is the solution of  for  fixed, .xo x y x– exp= y y 0

WL1

WL2

WL2

x

WL1

xo

y

x

WU1

ye
x–

xo WU2

y yx–

WL1
x WU1

– WL1
–

WU1

y
WL1

---------ln 1 y+ .ln= =

x

ye
x–

WU1
WL1

x WU1
– WL1

–

WL2

WL1
1 WU1

+ 

1 WL1
+

-----------------------------------=

ye
x–

WU2

y
WL2

---------ln 1 2y+
1 1 y+ ln+
------------------------------- .ln= =

WLi

WLi 1–
1 WUi 1–

+ 

1 WLi 1–
+

--------------------------------------------= WUi

y
WLi

---------ln = i 2 3   

y 0 y 1 e– 0 

WL1
y  y

1 y+
------------= WU1

y  1 y+ ln=

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  (31)

(32)

(33)

(34)

(35)

(36)

where

(37)

WL2
y  y 1 1 y+ ln+ 

1 2y+
---------------------------------------= WU2

y  1 2y+
1 1 y+ ln+
-------------------------------ln=

WL3
y 

y 1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

1 3y y 1 y+ ln+ +
--------------------------------------------------------------------------------------------------=

WU3
y  1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln=

WL4
y  =

y 1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+ 1 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+

1 4y y 1 2y+
1 1 y+ ln+
-------------------------------ln y 1 y+ ln 2 1 2y+

1 1 y+ ln+
-------------------------------ln++ + +

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

WU4
y  =

1 4y y 1 2y+
1 1 y+ ln+
-------------------------------ln y 1 y+ ln 2 1 2y+

1 1 y+ ln+
-------------------------------ln++ + +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+ 1 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ln

WL5
y 

yn5 y 

d5 y 
----------------= WU5

y 
d5 y 

n5 y 
-------------ln=

n5 y  1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+ 1 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+ =

1

1 4y y+ + 1 2y+
1 1 y+ ln+
-------------------------------ln y 1 y+ ln 2 1 2y+

1 1 y+ ln+
-------------------------------ln++

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+ 1 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ln+

https://doi.org/10.28924/ada/ma.2.14


 

  

(38)

3.1.2.  Notes. The approximation  is the approximation stated in
Equation 5 and is the basis for the Newton-Raphson iteration leading to Equation 23
and Equation 24. 

Consider the iteration formula specified in Equation 18 ([17], eqn. 12). With a starting
value of , it follows that a first order iteration leads to the approximation

(39)

which is  as specified by Equation 31.

3.1.3.  Results. The relative errors associated with the approximations, specified in
Theorem 3.1, are shown in Figure 10 and Figure 11, whilst the relative error bounds, for
the interval , are detailed in Table 1. Note the quadratic convergence.      

          

TABLE 1.  Relative error bounds, over the interval , for
the iterative approximations to the Lambert W function defined
in Theorem 3.1. 

Iteration 
order: i

Relative error 
bound for  

Relative error 
bound for  

1 increasing 0.381
2 increasing 0.0569
3

4

5

6

d5 y  1 5y y 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+ + +=

y 1 2y+
1 1 y+ ln+
------------------------------- 2 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+ln +

y 1 y+ 

3 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+ +

1 2y+
1 1 y+ ln+
------------------------------- 2 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+ln

ln

WU1
y  1 y+ ln=

x0 y 1 y+ =

W1 y  y
1 2y+
--------------- 1 1 y+ ln+  =

WL2
y 

0  

0  

WLi
WUi

8.32 10
3– 1.33 10

3–

3.88 10
6– 7.23 10

7–

1.08 10
12– 2.15 10

13–

9.33 10
26– 1.90 10

26–

https://doi.org/10.28924/ada/ma.2.14


 

  

3.2.  Alternative Iterative Formulas. The approximations  and  for the Lambert W
function, as specified in Theorem 3.1, can be also be defined via iterative formulas based
on numerator and denominator expressions.

Theorem 3.2.  Alternative Iterative Formulas for Lambert W. The approximations  and
, , for the Lambert W function can be specified according to

(40)

where

(41)

Proof. The proof is detailed in Appendix B.

FIGURE 10.  Graph of the relative errors in the iterative
approximations to the Lambert W function defined in Theorem 3.1.

y

WL1
y 

re y 

WL2
y 

WU1
y 

WU2
y 

WL3
y 

WU3
y 

WU4
y 

FIGURE 11.  Graph of the relative errors in the iterative
approximations to the Lambert W function defined in Theorem 3.1.

y

WL1
y re y 

WU1
y 

WL2
y 

WU2
y 

WL3
y 

WU3
y 

WL4
y 

WU4
y 

WLi
WUi

WLi

WUi
i 1 2   

WLi
y 

ni y 
di y 
------------= WUi

y  di y  ln
ni y 
y

------------ln– = i 1 2   

ni y  ni 1– y  1 di 1– y  ln
ni 1– y 

y
-------------------ln–+ = n0 y  y= d0 y  1=

di y  ni 1– y  di 1– y .+=

https://doi.org/10.28924/ada/ma.2.14


 

  
3.2.1.  Explicit Formulas. Explicit formulas, for the first to fourth order approximations,
are:

(42)

(43)

(44)

(45)

(46)

3.3.  Alternative Geometrical Approach. An alternative approach that leads to the
approximations , , , as specified in Theorem 3.1, is to utilize the transformation

, ,  in the relationship  which implies . The geometry
underpinning the iteration that leads to the approximations is illustrated in Figure 12.       

Theorem 3.3.  Alternative Geometrical Iteration. For  fixed, , iterative approximations
for the Lambert W function can be defined according to

(47)

and it is the case that  as specified in Theorem 3.1. 

Proof. Consider the case of  fixed and the geometry illustrated in Figure 12 which is
based on affine approximations to find, iteratively, the solution  to . The ini-

n1 y  y= d1 y  1 y+ =

n2 y  y 1 1 y+ ln+ = d2 y  1 2y+ =

n3 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ =

d3 y  1 3y y 1 y+ ln+ + =

n4 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ =

1 1 3y y 1 y+ ln+ + ln 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ln–+ ,

d4 y  1 3y y 1 y+ ln+ + y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ .+=

WU1
WU2



x z ln= x 0 z 1 y xe
x

= y z z ln=

FIGURE 12.   Illustration of the geometry underpinning the iterative relationship
to find approximations to  which is the solution of  for  fixed.zo z z ln y= y

1
z

y

zo

z z ln

z2z1

z 1–
z1 z1 ln z z1–  z1 ln 1+ +

y2

y1

z1 1 y+=

y y 0

zi zi 1–

zi 1– zi 1– ln y–

zi 1– ln 1+
-----------------------------------------– = z1 1 y+ =

xi zi ln =

xi WUi
y =

y

zo y z z ln=

https://doi.org/10.28924/ada/ma.2.14


 

  
tial value is established by the intersection of a first order Taylor series for  at the
point , which is , and the level . The solution is . A first order Taylor
series for  at this point is  and the intersection of this
approximation with the level  is the point  defined according to

(48)

Iteration in this manner leads to the stated general iteration formulas. 

3.3.1.  Explicit Formulas. Approximations, of orders one to three, are:

(49)

(50)

(51)

3.4.  Iterative Algorithm for (-1/e,0]. For the case of , the geometric
approach, illustrated in Figure 13, can be utilized to establish an algorithm for
determining approximations to the Lambert W function. 

Theorem 3.4.  Iterative Algorithm for (-1/e,0]. An iterative algorithm for defining
approximations to , for , is:

(52)

Proof. Consider the illustration shown in Figure 13. With an initial approximation for 

of , a first order Taylor series for , based on the point , with , is
. The intersection of this Taylor series with , at the point , leads to     

(53)

The value of  associated with  is . A first order Taylor series approxima-

tion for  at the point  is  and the intersection of this approxima-
tion with , at the point , leads to

z z ln

z 1= z 1– y z1 1 y+=

z z ln z1 z1 ln z z1–  z1 ln 1+ +

y z2

z2 z1

z1 z1 ln y–

z1 ln 1+
-----------------------------.–=

z1 y  1 y+ = x1 WU1
y  1 y+ ln = =

z2 y  1 2y+
1 1 y+ ln+
-------------------------------= x2 WU2

y  1 2y+
1 1 y+ ln+
-------------------------------ln = =

z3 y  1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------=

x3 WU3
y  1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

----------------------------------------------------------------------------------------------- .ln= =

y 1– e 0 

x W y = y 1 e– 0 

xi
yi 1– 1 xi 1–+ 

1 yi 1–+
-------------------------------------= x0 0= y0 y=

yi ye
xi–

.=

xo

x0 0= ye
x–

x0 y0  y0 y=

y0 1 x x0– –  x x1

x1

y0 1 x0+ 

1 y0+
-------------------------.=

ye
x–

x1 y1 ye
x1–

=

ye
x–

x1 y1  y1 1 x x1– – 

x x2

https://doi.org/10.28924/ada/ma.2.14


 

  (54)

The value of  associated with this value is . Iteration in this manner leads to
the general formula as stated in the theorem.

3.4.1.  Explicit Approximations. Approximations, for orders one to four, are:

(55)

(56)

3.4.2.  Results. The relative error in the iterative approximations specified in Theorem 3.4
are shown in Figure 14. For orders two, and higher, the approximation are more accurate
than the approximations detailed in Theorem 3.1.   

3.5.  Improved Approximations for [0,∞). Improved approximations, for the interval ,
can be established by using the iteration formula specified in Theorem 3.3 and by using

FIGURE 13.  Illustration of the geometry underpinning the iterative

relationship to find approximations to , the solution of , for the

case of  fixed and .

xo x ye
x–

=

y y 1 e– 0 

x1 x

y

x

x2

y1 ye
x– 1=

xo

y1 1 x x1– – 

ye
x–

y2 ye
x– 2=

y 1 x x0– – 

x0

x2

y1 1 x1+ 

1 y1+
-------------------------.=

ye
x–

y2 ye
x2–

=

WE1
y  y

1 y+
------------= WE2

y  y 1 2y+ 

1 y+  y e
y 1 y+ 

+ 
----------------------------------------------------=

WE3
y 

y y 2 3y+  1 y+ ey 1 y+ 
+

1 y+  y e
y 1 y+ 

+  y y 1 2y+ 

1 y+  y e
y 1 y+ 

+ 
----------------------------------------------------exp+

----------------------------------------------------------------------------------------------------------------------------------------=

0  

(57)WE4
y 

y

y
2

3 4y+  2y 1 y+ ey 1 y+ 
y 1 y+  y 1 2y+ 

1 y+  y e
y 1 y+ 

+ 
----------------------------------------------------exp+ + +

1 y+  y 1 3y e
y 1 y+ 

+ + 

1 y+  y e
y 1 y+ 

+ 
----------------------------------------------------exp

1 y+  y e
y 1 y+ 

+  y
y 1 2y+ 

1 y+  y e
y 1 y+ 

+ 
----------------------------------------------------exp+ 

y
y y 2 3y+  1 y+ ey 1 y+ 

+ 

1 y+  y e
y 1 y+ 

+  y
y 1 2y+ 

1 y+  y e
y 1 y+ 

+ 
----------------------------------------------------exp+

----------------------------------------------------------------------------------------------------------------------------------------exp+

----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------=

https://doi.org/10.28924/ada/ma.2.14


 

  an initial approximation of  rather than . The resulting
approximations, of orders one to four, are:

(58)

(59)

(60)

(61)

The improvement in the relative error bounds for the interval , for values of 
close to optimum, are detailed in Table 2. 

For the case of a third order approximation, and for , the maximum relative

error is . Thus, the approximation

(62)

FIGURE 14.   Graph of the relative error in approximations to , for the interval

, based on the iterative algorithms specified in Theorem 3.4 and Theorem 3.1.

W y 

1 e– 0 

y

WE2
y 

re y 

WE4
y 

WE1
y  WL1

y 

WE3
y 

WL2
y 

WU1
y 

WU2
y 

WL3
y  WU3

y 

WU4
y 

WL4
y 

z1 y  1 ky+= z1 y  1 y+=

z1 y  1 ky+ = Wk1
y  1 ky+ ln =

z2 y  1 1 k+ y+
1 1 ky+ ln+
----------------------------------= Wk2

y  1 1 k+ y+
1 1 ky+ ln+
----------------------------------ln =

z3 y  1 2 k+ y y 1 ky+ ln+ +

1 1 ky+ ln+  1 1 1 k+ y+
1 1 ky+ ln+
----------------------------------ln+

-----------------------------------------------------------------------------------------------------=

Wk3
y  z3 y  ln =

z4 y  =

1 3 k+ y y 1 1 k+ y+
1 1 ky+ ln+
----------------------------------ln y 1 ky+ ln 2 1 1 k+ y+

1 1 ky+ ln+
----------------------------------ln++ + +

1 1 ky+ ln+  1 1 1 k+ y+
1 1 ky+ ln+
----------------------------------ln+ 1 1 2 k+ y y 1 ky+ ln+ +

1 1 ky+ ln+  1 1 1 k+ y+
1 1 ky+ ln+
----------------------------------ln+

-----------------------------------------------------------------------------------------------------ln+

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Wk4
y  z4 y  .ln=

0   k

k 4465
10000
---------------=

1.023 10
4–

Wk3
y  1 2 k+ y y 1 ky+ ln+ +

1 1 ky+ ln+  1 1 1 k+ y+
1 1 ky+ ln+
----------------------------------ln+

-----------------------------------------------------------------------------------------------------ln = k 4465
10000
---------------=

https://doi.org/10.28924/ada/ma.2.14


1 0.381 0.570 0.518 0.465

 

  

represents a good compromise between accuracy and complexity for the interval 
with a relative error bound of close to .     

3.6.  Higher Accuracy via Iterative Quadratic Approximations. The iterative
approximation detailed in Theorem 3.1 can be improved upon by utilizing quadratic,
rather than affine, approximations as illustrated in Figure 15.      

Theorem 3.5.  Iterative Quadratic Approximations for Lambert W Function. An iterative
formula, based on quadratic approximations, for the Lambert W function, and valid for

, is

(63)

Proof. The proof is detailed in Appendix C.

3.6.1.  Explicit Approximations. Approximations, of orders one to three, are:

(64)

TABLE 2.  Relative error bounds, over the interval , in the approximations , , to

the Lambert W function.

Iteration 
order: i 

Relative error 
bound: k = 1

Relative error 
bound: k = 0.4

Relative error 
bound: k = 0.45

Relative error 
bound: k = 0.5

2 0.0569 0.0335 0.0250 0.0184
3

4

5

6

0  

10
4–

0   Wki
y  i 1 2  6   

1.33 10
3– 1.87 10

4– 1.05 10
4– 1.50 10

4–

7.23 10
7– 6.46 10

9– 4.96 10
9– 9.97 10

9–

2.15 10
13– 7.98 10

18– 1.11 10
17– 4.43 10

17–

1.90 10
26– 1.24 10

35– 5.53 10
35– 8.78 10

34–

FIGURE 15.   Illustration of the geometry underpinning an iterative relationship,
based on quadratic approximations, to find an approximation to  which is the

solution of  for  fixed, .

xo
x y x– exp= y y 0

WL1

WL2

WL2

x

WL1

xo

x

WU1

ye
x–

xo WU2

y 1 x– x
2

2
-----+ WL1

x WU1
– WL1

x WU1
– 2

2
--------------------------WL1

+–

y

y 1 e–

WLi
1

WLi 1–

-------------- 1 WLi 1–
1 WUi 1–

+  1 WLi 1–

2
– 2WLi 1–

1 WUi 1–
+ +–+ =

WUi

y
WLi

---------ln = WL1

y
1 y+
------------= WU1

1 y+ .ln=

WL1
y  y

1 y+
------------= WU1

y  1 y+ ln=

https://doi.org/10.28924/ada/ma.2.14


 

  

(65)

(66)

(67)

3.6.2.  Results. The relative errors in the approximations specified in Theorem 3.5 are
shown in Figure 16 and Figure 17. The relative error bounds, for the interval , are
detailed in Table 3 and the convergence is cubic in nature. The approximation  has a

relative error bound for the interval  of .           

4.  CONVERGENCE

Consider the case of , , and the error definitions

(68)

associated with the  upper and lower approximations detailed in Theorem 3.1 and as
illustrated in Figure 18. The following results hold:       

TABLE 3.   Relative error bounds, over the interval , for
the approximations to the Lambert W function detailed in
Theorem 3.5.

Iteration 
order i 

Maximum relative 
error in  

Maximum relative 
error in  

1 increasing 0.381
2 increasing 0.0122
3

4

5

WL2
y  1

y
--- 1 2y y 1 y+ ln 1 4y 2y

2
2y 1 y+  1 y+ ln+ + +–+ +=

WU2
y  y

2

1 2y y 1 y+ ln 1 4y 2y
2

2y 1 y+  1 y+ ln+ + +–+ +

-----------------------------------------------------------------------------------------------------------------------------------------------ln=

WL3
y  1

1 2y y 1 y+ ln q2 y –+ +
------------------------------------------------------------------ =

y 1 2y y 1 y+ ln q2 y –+ + 1 y
2

1 2y y 1 y+ ln q2 y –+ +
------------------------------------------------------------------ln+ –+

y
2

1 2y y 1 y+ ln q2 y –+ + 2– +

2y 1 2y y 1 y+ ln q2 y –+ + 1 y
2

1 2y y 1 y+ ln q2 y –+ +
------------------------------------------------------------------ln+

q2 y  1 4y 2y
2

2y 1 y+  1 y+ ln+ + +=

WU3
y  y

WL3
i 

---------------- .ln=

0  
WU3

0   1.26 10
6–

0  

WLi
WUi

1.02 10
5– 1.26 10

6–

1.66 10
17– 2.04 10

18–

7.88 10
53– 9.63 10

54–

y 0 xo W y =

Li xo WLi
– = Ui

WUi
xo– =

ith

https://doi.org/10.28924/ada/ma.2.14


 

  

Theorem 4.1.  Convergence of Iterative Algorithm. For the case of  fixed and ,
the errors associated with the lower approximations , , detailed in
Theorem 3.1, are

(69)

FIGURE 16.   Graph of the relative errors in approximations to
 as specified in Theorem 3.5.W y 

y

WL2
y 

re y 

WL1
y 

WU1
y 

WU2
y 

WU3
y 

FIGURE 17.  Graph of the relative errors in approximations to  as

specified in Theorem 3.5.

W y 

y

WL1
y re y 

WL2
y 

WU1
y 

WU2
y 

WL3
y 

WU3
y 

FIGURE 18.   Error definitions associated with the  upper and lower
approximations to .

ith
xo W y =

WLi
WLi 1+

x

xo

y

x

WUi

ye
x–

xo WUi 1+

UiLi

y xo W y =

WLi
i 1 2   

Li Li 1–

WLi 1–
WUi 1–

WLi 1–
– 

1 WLi 1–
+

-------------------------------------------------------– Li 1–

WLi 1–
Li 1–

Ui 1–
+ 

1 WLi 1–
+

---------------------------------------------------–= = L1
xo

y
1 y+
------------– .=

https://doi.org/10.28924/ada/ma.2.14


 

  The following results hold: First, the sequence  is a monotonically increasing
sequence, i.e. . Second, the errors , , define a monotonically
decreasing sequence, i.e. . Third, convergence is guaranteed, i.e. ,

 and .

Proof. The proof of these results is detailed in Appendix D.

5.  SPLINE BASED APPROXIMATION FOR [-1/e,0]

The approximations, detailed above in Theorem 3.1 and Theorem 3.4, are sharp at the
origin but not at the point . It is useful to have approximations that are sharp at both
points and which converge throughout the interval , e. g. [3], eqn. 7 and [17], eqn.
20. The latter approximation is sharp at  but not at the origin. One approach, with
potential, is to utilize the two point spline approximation for a function  as specified by
Howard, [16], eqn. 40. For an interval , the  order approximation can be written
in the form (see Appendix E)

(70)

where

(71)

5.1.  Spline Based Approximations for [-1/e,0]. The spline approximation detailed in
Equation 70 has the potential to provide approximations for the Lambert W function in
the interval  with exact values at the end points of this interval. However, an
initial problem is that the derivatives of the Lambert W function are undefined at the
point . This problem can be overcome by utilizing two suitable transformations. 

Lemma 1.  Transformations. With , , the first transformation

(72)

with ,  and , ,  yields

(73)

(74)

WLi

WLi
WLi 1–

 Li i 1 2   

Li 1+
Li Lii 

lim 0=

WLii 
lim xo= WUii 

lim xo=

1 e–

1– e 0 
1– e

f

   nth

fn x   x– n 1+
an r x – r

r 0=

n

 x – n 1+
bn r  x– r

r 0=

n

+ = x   ,

an r
1

 – n 1+
---------------------------- f

r u–   
r u– !

----------------------- n u+ !
u!n!

-------------------
u 0=

r

 1

 – u
--------------------  =

bn r
1

 – n 1+
---------------------------- 1– r u–

f
r u–   

r u– !
-------------------------------------------- n u+ !

u!n!
-------------------

u 0=

r

 1

 – u
--------------------. =

1 e– 0 

1 e–

f x  xe
x

= x 1–

y1 g1 x1  1
e
--- f x1 1– + ==

x x1 1–= x1 0 y y1
1
e
---–= y 1–

e
------ y1 0

g1 x1  1
e
--- 1 x1 1– e

x1+  = x1 0 

W y  f
1–
y  g1

1–
y 1

e
---+ 1– = y 1–

e
------.=

https://doi.org/10.28924/ada/ma.2.14


 

  
With the second transformation of , it follows that

(75)

(76)

Proof. The proofs for these results are detailed in Appendix F.

5.1.1.  Graphs and Values. Consistent with Equation 76, the Lambert W function is
defined in terms of  for . Relevant values associated with the points

 and  are specified in Table 4. The graph of , and its inverse ,
are shown in Figure 19.               

5.1.2.  Spline Based Approximations. Consistent with Equation 76, and the values
tabulated in Table 4, an approximation for the Lambert , over the interval ,
requires an approximation to the inverse of , , i.e. , , to

be determined. A spline approximation for , of order , requires derivatives, of orders
zero to , at the points  and  to be determined. Such values can be determined
from the derivatives of  at the points  and  as detailed in Appendix G. The following
approximations result.

Theorem 5.1.  Spline Based Approximations for the Lambert W Function. The  order
spline based approximation for the Lambert W function, based on the transformation 
and the points  and , is

(77)

TABLE 4.  Values associated with the points  and . 

g x1  g1 x1 =

g x1  1
e
--- 1 x1 1– e

x1+  = x1 0 

W y  g
1–

y 1
e
---+ 1– = y 1

e
---– .

g
1–

y 1 e– 0 

y 1 e–= y 0= g x1  g
1–
y2 

FIGURE 19.  Graph of  and its inverse . y2 g x1 = g
1–
y2 

x1 y2

g
1–
y2 

g x1 

1 e

y 1 e–= y 0=

y y1 y 1 e+= x W y = x1 x 1+= y2 g x1 =

1– e 0 1– 0 0

0 1 e 0 1 1 e

W 1 e– 0 

g x1  x1 0 1  g
1–
y2  y2 0 1 e 

g
1–

n

n 0 1 e
g 0 1

kth

g

1 e–  0

W k y  1– 2e y 1
e
---+ 1 1 y 1

e
---+ 2 y 1

e
---+ 3 y 1

e
---+

3 2
+ + + + 2k y 1

e
---+

k
++ =

https://doi.org/10.28924/ada/ma.2.14


 

  for  and for appropriately defined constants. 

Proof. The proof is detailed in Appendix G. 

5.1.3.  Explicit Approximations. Explicit approximations, of orders one to four, are:

(78)

(79)

(80)

(81)

k 1 2   

W 1 y  1– 2e y 1
e
---+ 1 2 e 1 1

2e
---------- 3

2 2
----------–+ y 1

e
---+– e 1 2– 2

e
-------+ y 1

e
---+++=

W 2 y  1– 2e y 1
e
---+

1 2e
3

---------- y 1
e
---+– 6e 2 1–  7

2
-------– 2 2

e
----------– y 1

e
---+ –+

e
3 2 17

2
------- 8– 6 2e– 4 2

e
----------– y 1

e
---+

3 2
+

e
2 10 2

3
------------- 3– 5e

2
-------– 2 2– y 1

e
---+

2

+=

W 3 y  1– 2e y 1
e
---+

1 2e
3

---------- y 1
e
---+– 11e

36
--------- y 1

e
---+ –+

e
3 2 191

9
--------- 125

3 2
----------– 25 e

2
------------- 8 2

e
---------- 6 2

e
3 2

----------+ + + y 1
e
---+

3 2
+

e
2 281

6
--------- 146 2

3
----------------– 32 2e 22 2 18 2

e
-------------+ + + y 1

e
---+

2
–

e
5 2 335

9
--------- 40 2– 55e

3 2

2
----------------- 20 2 e 18 2

e
-------------+ + + y 1

e
---+

5 2
+

e
3 371

36
--------- 34 2

3
-------------– 8 2e

2
6 2e 6 2+ + + y 1

e
---+

3

+=

W 4 y  1– +=

2e y 1
e
---+

1 2e
3

---------- y 1
e
---+– 11e

36
--------- y 1

e
---+ 43e

3 2

135 2
----------------- y 1

e
---+

3 2
–+ +

e
2 4075

27 2
------------- 895

12
---------– 91e

2
---------– 61

2
-------– 27 2

e
-------------– 64 2

3e
2

-------------– y 1
e
---+

2
–

e
5 2 6658 2

27
------------------- 2126

9
------------– 315e

3 2

2
--------------------– 110 2e– 102 2

e
----------------– 256 2

3e
3 2

----------------– y 1
e
---+

5 2
+

e
3 8467 2

27
------------------- 1175

4
------------– 207 2e

2
– 149 2e– 144 2– 128 2

e
----------------– y 1

e
---+

3
–

e
7 2 4904 2

27
------------------- 502

3
---------– 245e

5 2

2
--------------------– 90 2e

3 2
– 90 2e– 256 2

3 e
----------------– y 1

e
---+

7 2
+

e
4 10843

135 2
---------------- 1315

36
------------– 55e

3

2
-----------– 41e

2

2
-----------– 21 2e– 64 2

3
-------------– y 1

e
---+

4



https://doi.org/10.28924/ada/ma.2.14


 

  
5.1.4.  Results. The relative errors in the spline based approximations to the Lambert W
function, as specified in Theorem 5.1, of orders one to five, are shown in Figure 20. The
relative error bounds over the interval , for first to fifth order approximations,
respectively, are: , , ,  and . By
construction, the approximations are sharp at the points  and . The results shown
in Figure 20 indicate that the sequence of approximations have good convergence to the
Lambert W function over the interval  and modest convergence over the interval

.     

6.  IMPROVED APPROXIMATIONS VIA ITERATION

Iteration is, potentially, effective in improving the accuracy of an initial approximation.
One potential approach is to utilize the iteration potential in the fundamental relation-
ship for the Lambert W function as specified by Equation 116. An alternative approach is
to utilized the Newton-Raphson method. These two approaches are detailed below.

6.1.  Inherent Iteration. The basis for an iterative relationship for the Lambert W function
is Equation 116, i.e.    

(82)

It then follows that an approximation, , for , can potentially be improved upon
according to

(83)

(84)

1 e– 0 

1.27 10
2– 9.68 10

4– 8.69 10
5– 8.46 10

6– 8.65 10
7–

1 e– 0

1 e– 0 
0 1 

FIGURE 20.   Graph of the relative errors in the spline based approximations,
of orders one to five, to the Lambert W function as specified in Theorem 5.1.

y

order 1

order 5

re y 

1 e–

W y  y ln W y  ln– .=

W0 y  W y 

W1 y  y ln W0 y  ln–=

W2 y  y ln y ln W0 y  ln–ln–=

W21 y  y ln y ln W1 y  ln–ln–=

https://doi.org/10.28924/ada/ma.2.14


 

  

(85)

(86)

etc. The number of possible permutations for approximations to the Lambert W function is
clearly large as the iteration order increases. Representative relative error bounds, for
the interval , are tabulated in Table 5 for the base approximations of

,  and  as specified in Theorem 3.1. Graphs
of the relative errors, based on the approximations  and ,
are shown, respectively, in Figure 21 and Figure 22.              

W3 y  y ln y ln y ln W0 y  ln–ln–ln–=

W31 y  y ln y ln y ln W1 y  ln–ln–ln–=

W32 y  y ln y ln y ln W2 y  ln–ln–ln–=

W3 21 y  y ln y ln y ln W21 y  ln–ln–ln–=

W4 y  y ln y ln y ln y ln W0 y  ln–ln–ln–ln–=

0 
W0 y  WU3

y = W0 y  WU4
y = W0 y  WU5

y =

W0 y  WU4
y = W0 y  WU5

y =

FIGURE 21.   Graph of the relative errors in approximations to ,

based on  (Equation 35).

W y 

W0 y  WU4
y =

y

W0

W1

W3 21
W32 W4

W2

W3

W31

W21

re y 

FIGURE 22.   Graph of the relative errors in approximations to ,

based on  (Equation 36).

W y 

W0 y  WU5
y =

y

W0

W1

W3 21 W32

W2

W3

W31 W21

re y 

W4

https://doi.org/10.28924/ada/ma.2.14


 

  

   

6.1.1.  Explicit Approximations. The approximation , based on , is

(87)

and has a maximum relative error bound, over the interval , of  which is a
factor of  lower that the original approximation  whose relative error bound is

.
The approximation , based on , is

(88)

where  is specified by Equation 35. The maximum relative error bound, over the

interval , is  which is a factor of  lower that the original approximation
 whose relative error bound is .

The approximation , based on , is

TABLE 5.  Relative error bounds, for the interval , based on iteration
and for the specified base approximations of  (Equation 33), 

(Equation 35) and  (Equation 36).

Iteration 
Form

0 
WU3

y  WU4
y 

WU5
y 

W0 y  WU3
y = W0 y  WU4

y = W0 y  WU5
y =

W0 1.33 10
3– 7.23 10

7– 2.15 10
13–

W1 4.08 10
4– 1.84 10

7– 4.95 10
14–

W2 1.97 10
4– 6.03 10

8– 1.31 10
14–

W21 1.40 10
4– 2.47 10

8– 3.95 10
15–

W3 1.40 10
4– 2.47 10

8– 3.95 10
15–

W31 1.46 10
4– 1.23 10

8– 1.34 10
15–

W32 2.33 10
4– 7.43 10

9– 5.04 10
16–

W3 21 6.65 10
4– 5.36 10

9– 2.11 10
16–

W4 1.46 10
4– 1.23 10

8– 1.34 10
15–

W3 W0 y  WU3
y =

W3 3 y  y ln y ln y ln WU3
y  ln–ln–ln–=

y ln y ln y ln 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------lnln–ln–ln–=

0  1.40 10
4–

9.5 WU3
y 

1.33 10
3–

W2 W0 y  WU4
y =

W2 4 y  y ln y ln WU4
y  ln–ln– =

WU4
y 

0  6.03 10
8– 12

WU4
y  7.23 10

7–

W3 W0 y  WU4
y =

https://doi.org/10.28924/ada/ma.2.14


 

  
(89)

and has a maximum relative error bound, over the interval , of  which is a
factor of  lower that the original approximation  whose relative error bound is

.

6.2.  Newton-Raphson Iteration. Consistent with Equation 22, the approximations stated
in Theorem 3.1, Theorem 3.2 and Theorem 3.5 can be utilized as the basis for iterative
approximations based on the Newton-Raphson method. Results are detailed in Table 6. 
    

As an example, the approximation arising from a first order iteration of 

(Equation 33), has a maximum relative error bound, over the interval , of 
which is a factor of  lower that the original approximation whose relative error bound
is . The approximation is:

(90)

TABLE 6.   Relative error bounds, for the interval , based on Newton-Raphson iteration
and for the specified base approximations, , of , ,  and .

Iteration 
Order (Equation 31) (Equation 33) (Equation 35) (Equation 36)

W3 4 y  y ln y ln y ln WU4
y  ln–ln–ln– =

0  2.47 10
8–

29.3 WU4
y 

7.23 10
7–

0 
W0 y  WU2

y  WU3
y  WU4

y  WU5
y 

W0 y  WU2
y = W0 y  WU3

y = W0 y  WU4
y = W0 y  WU5

y =

0 5.69 10
2– 1.33 10

3– 7.23 10
7– 2.15 10

13–
1 8.28 10

3– 5.12 10
6– 1.49 10

12– 1.30 10
25–

2 3.48 10
4– 9.61 10

11– 6.98 10
24– 5.02 10

50–
3 9.95 10

7– 3.91 10
20– 1.62 10

46– 7.67 10
99–

4 8.21 10
12– 7.08 10

39– 9.04 10
92– 1.81 10

196–
5 5.60 10

22– 2.43 10
76– 2.85 10

182– 1.02 10
391–

WU3
y 

0  5.12 10
6–

260

1.33 10
3–

W1 3 y  1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln –=

1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln

y 1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

1 3y y 1 y+ ln+ + 
--------------------------------------------------------------------------------------------------–

1 1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln+

-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

https://doi.org/10.28924/ada/ma.2.14


 

  7.  APPLICATIONS
7.1.  Approximation with Fixed Relative Error Bound. There are many applications
where the Lambert W function is used to model the physical nature/characteristics of an
entity which are positive in nature. In many of these cases the parameter values are not
known with high accuracy. For such cases, highly accurate computation of the Lambert W
function is not required and a fixed approximation, with a set relative error bound, is
useful rather than relying on, for example, iterative approximations where the relative
error achieved depends on the initial approximate used. The relatively simple
approximation for the Lambert W function, as specified by Equation 33, i.e. 

(91)

with a relative error bound of  over the interval , is likely to be useful.
One application, for example, is in the evaluation of the collector current in a common
emitter circuit, e.g. [2], eqn. 21.

For the interval , the approximation  detailed in Equation 79, is of
modest complexity, is sharp at the points  and  and has a relative error bound of

. 
For highly accurate approximations over the interval , an explicit analytical approx-
imation, with a relative error bound of , can be specified by utilizing the fifth
order iterative approximation, denoted , arising from the iteration specified by
Equation 47 and with ,  (see Table 2).

An alternative analytical expression, with a relative error bound of  over
the interval , is

(92)

where  and  are defined, respectively, in Equation 37 and Equation 38. This expres-
sion arises from a first iteration of the Newton-Raphson method (Equation 22) utilizing
the fifth order approximation  specified in Equation 36. The relative error bound is
specified in Table 6. 

7.2.  Upper/Lower Bounds for Lambert W. There is interest in upper/lower bounds for
the Lambert W function, e.g. [15] and [24]. Alzahrani and Salem, [1], detail bounds for

. The following bounds were proposed by Hoorfar (see, [17], eqn. 21)

(93)

For the interval , the relative error bound associated with the lower bounded func-
tion is ; the relative error bound for the upper bounded function is .

WU3
y  1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

-----------------------------------------------------------------------------------------------ln =

1.33 10 3–
0 

1 e– 0  W 2 y 
1 e– 0

9.68 10
4–

0 

7.98 10
18–

Wk5

z1 1 ky+= k 0.4=

1.30 10
25–

0 

W1 5 y 
d5 y 
n5 y 
-------------ln

d5 y 
n5 y 
-------------ln

yn5 y 
d5 y 

----------------–

1
d5 y 
n5 y 
-------------ln+

-----------------------------------------------– =

n5 d5

WU5
y 

W 1–

y
y ln

-------------ln y ln ln
2 y ln

-----------------------+ W y  y
y ln

-------------ln e
e 1–
----------- y ln ln

y ln
-----------------------.+ 

e  
0.0568 0.207

https://doi.org/10.28924/ada/ma.2.14


 

  
The bounds proposed in [17], eqn. 25, 27, have modest relative errors for  and are:

(94)

The relative error bounds in the lower and upper bounded functions, over the interval
, respectively, are  and . The relative errors decrease for .

By construction, , , as defined in Theorem 3.1, are a sequence of
increasingly accurate lower bounds for . Similarly, , , is a
sequence of increasingly accurate upper bounds for . For example:

(95)

with relative error bounds for the interval  of, respectively  and 
(see Table 1). Higher order approximations lead to lower relative bounds and these can
be made arbitrarily small. The relative error bounds associated with 

over the interval  are, respectively,  and . The relative error
bounds associated with  over the interval  are, respectively,

 and .
One potential application for the upper bound is a bound for the prime counting func-

tion, e.g. [27].

7.3.  Spline Approximations Based on Upper/Lower Bounds. Consider the  upper,
, and lower, , bounded functions for the Lambert W function as illustrated in

Figure 23 and as defined in Theorem 3.1. For  fixed at , a spline approximation, as
specified by Equation 70 and based on the points ,  and

, , can readily be determined. From such an approximation, an
approximation to  can then be specified.     

Theorem 7.1.  Spline Approximations Based on Upper/Lower Bounds. Consider the 
lower and upper bounded approximations,  and , defined in Theorem 3.1. The zero

y e

y
y ln

-------------ln

y
y ln

-------------ln

1 y
y ln

-------------ln+

---------------------------------– 1 y ln ln
y ln

-----------------------–ln W y  

y
y ln

-------------ln 1 y ln ln
y ln

-----------------------– 1

1 y ln ln
y ln

-----------------------–ln

1 y
y ln

-------------ln+

-------------------------------------------– .ln–

e  5.96 10
3– 4.10 10

3– y 10»

WLi
y  i 1 2   

W y  WUi
y  i 1 2   

W y 

WL3
y 

y 1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

1 3y y 1 y+ ln+ +
--------------------------------------------------------------------------------------------------= W y  

WU3
y  1 3y y 1 y+ ln+ +

1 1 y+ ln+  1 1 2y+
1 1 y+ ln+
-------------------------------ln+

----------------------------------------------------------------------------------------------- ln= y 0 

0  8.32 10
3– 1.33 10

3–

WL4
y  W y  WU4

y  

0  3.88 10
6– 7.23 10

7–
WL5

y  W y  WU5
y   0 

1.08 10
12– 2.15 10

13–

ith

WUi
WLi

y yo

uo uo exp uo  uo WLi
yo =

vo vo exp vo  vo WUi
yo =

xo W yo =

ith

WLi
WUi

https://doi.org/10.28924/ada/ma.2.14


 

  
order spline approximation for the Lambert W function, based on the  approximations

 and , is    

(96)

The  order spline approximation for the Lambert W function, based on the  approx-
imations  and , is

(97)

where ,  and  is defined by Equation 120. 

Proof. The proof is detailed in Appendix H.

7.3.1.  Results. Results are detailed in Table 7 and clearly show the high accuracy of the
approximations. For example, the zero order spline approximations, as specified by
Equation 96, yields relative error bounds, for the interval , of , 
and , respectively, based on third, fourth and fifth order approximations for the

FIGURE 23.   Illustration of upper and lower bounded approximations to the Lambert W
function and the points , , which are the basis for spline

based approximations. 

uo uo exp uo  vo vo exp vo 

uo uo exp

uo WLi
yo =

y

xo W yo =
WLi

y 

W y 

x
spline approx. based on WUi

y 

vo WUi
yo =

vo vo expyo

ith

WLi
WUi

W0 i y 
WLi

y WUi
y  WUi

y  exp WLi
y  exp– y WUi

y  WLi
y – +

WUi
y  WUi

y  exp WLi
y  WLi

y  exp–
-----------------------------------------------------------------------------------------------------------------------------------------------------------------------------.=

nth ith

WLi
WUi

Wn i y 
vo vo exp y– n 1+

vo vo exp uo uo exp– n 1+
------------------------------------------------------------------------ =

y uo uo exp– r

n r+ !uo

r!n! vo vo exp uo uo exp– r
-------------------------------------------------------------------------- +

pr u– uo e
r u– uo–

r u– ! 1 uo+ 2 r u–  1–
------------------------------------------------------------- n u+ !

u!n!
-------------------

u 0=

r 1–

 1

vo vo exp uo uo exp– u
-----------------------------------------------------------------


r 0=

n

 +

y uo uo exp– n 1+

vo vo exp uo uo exp– n 1+
------------------------------------------------------------------------ 

vo vo exp y– r

n r+ !vo

r!n! vo vo exp uo uo exp– r
-------------------------------------------------------------------------- +

1– r u–
pr u– vo e

r u– vo–

r u– ! 1 vo+ 2 r u–  1–
------------------------------------------------------------------ n u+ !

u!n!
-------------------

u 0=

r 1–

 1

vo vo exp uo uo exp– u
-----------------------------------------------------------------


r 0=

n



uo WLi
y = vo WUi

y = pk

0   3.84 10
5– 8.56 10

12–

6.80 10
25–

https://doi.org/10.28924/ada/ma.2.14


 

  
Lambert W function detailed in Equation 3.1 Such convergence is approximately
quadratic.

7.3.2.  Application. The Omega constant, defined as , can be evaluated by using
Equation 96 with relative errors, respectively, of , ,  and

 for the case of upper and lower bounded approximations of orders two to five,
i.e. , ,  and .       

7.4.  Asymptotic Approximations. As is evident in the results shown in Figure 11, apart
from the results for  and , the relative errors in the approximations defined in
Theorem 3.1, for a set order, decrease as their argument increases, i.e. for a set order, the
approximations asymptotically approach the Lambert W function as their arguments
become unbounded. Thus:

(98)

7.4.1.  Lambert W Function and Prime Counting Function. The prime number theorem
states that the relative error between the prime counting function  and 
decreases to zero as , i.e.

(99)

TABLE 7.  Relative error bounds, over the interval , for spline approximations to the
Lambert W function based on upper and lower bounded functions. 

Upper/lower bounded 
functions

Spline 
order Approximation

Relative error 
bound 

,  1 increasing re as 

   (Equation 31) 2 increasing re as 

,  0

   (Equation 32, Equation 33) 1

2

3

4

,  0

   (Equation 34, Equation 35) 1

2

3

4

,  0

   (Equation 36) 1

W 1 

1.94 10
5– 4.71 10

11– 2.76 10
22–

9.48 10
45–

W0 2 1  W0 3 1  W0 4 1  W0 5 1 

0  

WL2
y  WU2

y  W1 2 y 

W2 2 y 

WL3
y  WU3

y  W0 3 3.84 10
5–

W1 3 1.92 10
8–

W2 3 1.46 10
11–

W3 3 1.31 10
14–

W4 3 1.27 10
17–

WL4
y  WU4

y  W0 4 8.56 10
12–

W1 4 5.60 10
22–

W2 4 5.05 10
32–

W3 4 5.18 10
42–

W4 4 5.68 10
52–

WL5
y  WU5

y  W0 5 6.80 10
25–

W1 5 3.01 10
48–

WL1
y  WL2

y 

WLi
y  W y  i 3   

WUi
y  W y  i 1 2 3    .

 y  y y ln
y 

 y  y y .ln

https://doi.org/10.28924/ada/ma.2.14


 

  

As  (Equation 115), an equivalent statement for the prime number theo-
rem is

(100)

With the manipulation of 

(101)

and with  for  large, it follows that

(102)

which has been proposed by Visser [27] and briefly discussed by Iacono and Boyd, [17],
section 4.4. Visser has proved that  is an upper bound for the prime counting func-
tion whilst  is a lower bound for  large. The magnitude of the relative error in the
approximation of  is lower than the magnitude of the relative error in the
approximation  for  greater than around  with the relative error
decreasing as  increases. However, the magnitude of the relative error in the approxi-
mation  is of the order of  for .

7.5.  Floor and Integral of Floor of Lambert W. It is possible to specify approximations
for the Lambert W function, with a set accuracy bound, if an explicit expression for the
floor of the Lambert function can be specified. The graph of  is shown in
Figure 24.      

Theorem 7.2.  Floor of Lambert W. The floor of the Lambert W function can be ascertained,
without knowledge of the function itself, according to

(103)

where,  is fixed,  is the unit step function and  is an upper bound for
the Lambert W function as specified in Theorem 3.1.

W z z ln  z ln=

 y  y W y y ln .

W y y ln  y ln xe
x ln x x ln+ W y  W y  ln+ = = = =

W y  W y  ln» y

 y  y W y  

y W y 
y y ln y

 y  y W y 
 y  y W y y ln  y 5000

y

 y  y W y  0.05 y 10
9

=

W y 

FIGURE 24.   Graph of  and . W y  W y 

y

W y 
W y 

2e
2e 4e

4
3e

3

W y  u y ke
k

– 
k 1=



 u y ke
k

– 
k 1=

1 WUi
y +

 = = y 0 

i 1 2    u WUi
y 

https://doi.org/10.28924/ada/ma.2.14


 

  

Proof. This result follows from the fact that  and , ,  fixed, is
an upper bound for , i.e. . It then follows that the upper limit of the
summation can be specified as .   

7.5.1.  Notes. The simplest upper bound, specified in Theorem 3.1, for the Lambert W
function is  and, thus:

(104)

The graphs of , for , are shown in Figure 25 and it follows
that the use of  as the upper limit for the summation results in
an increasing small number of additional zero terms in the summation defining  as
 increases. The use of  results, depending on the value of , in an additional

zero term in the summation defining .    

7.5.2.  Integral of Floor of Lambert W. Using the result for the floor of the Lambert W
function, as specified in Theorem 7.2, it is possible to explicitly specify the integral of the
floor of the Lambert W function.

Theorem 7.3.  Integral of Floor of Lambert W. The integral of the floor of the Lambert W
function can be explicitly specified according to

(105)

where  is specified in Theorem 7.2.

Proof. The required result follows, consistent with the graph of  shown in
Figure 24, according to

W ke
k  k= WUi

y  i 1 2    i

W y  W y  WUi
y 

1 WUi
y +

WU1
y  1 y+ ln=

W y  u y ke
k

– 
k 1=

1 1 y+ ln+

 = y 0 .

1 WUi
y  W y –+ i 1 2 3  

1 WU1
y + 1 1 y+ ln+=

W y 
y 1 WU2

y + y

W y 

FIGURE 25.   Graph of  for . 1 WUi
y  W y –+ i 1 2 3 

y

1 WUi
y  W y –+

i 1=

i 2=

i 3=

W   d
0

y


e

e 1– 2
-------------------

1– e
W y 

1 W y  2 W y  2
–+ + +

W y  1– W y + e1 W y +
W y  2

e
W y  1–

+
 +=

W y  y W y  e
W y 

– 

W y 

W y 

https://doi.org/10.28924/ada/ma.2.14


 

  

(106)

where the following result has been used: 

(107)

7.6.  Set Accuracy Approximation for Lambert W. Consider a set accuracy limit of 
required for the evaluation of . This can be achieved by a step approximation, with a
resolution of , and such an approximation is:

(108)

where , ,  fixed, is a set function defined in Theorem 3.1. As an example,
the error in the approximation to , with a resolution of , is shown in
Figure 26.      

7.6.1.  Computationally Efficient Implementation. The direct approximation detailed in
Equation 108, for a set error level of , requires, approximately, a summation of

 terms. The approach detailed below requires, approximately, the summation of
 terms which represents a significant reduction for  modest to large. For

example, for  and , the direct approach requires a summation of
approximately  terms whilst the approach detailed below requires close to  terms. 

W   d
0

y

 k k 1+ ek 1+
ke

k
– 

k 0=

max 0 W y  1– 

 W y  y W y  e
W y 

– +=

e

e 1– 2
-------------------

1– e
W y 

1 W y  2 W y  2
–+ + +

W y  1– W y + e1 W y +
W y  2

e
W y  1–

+
 +=

W y  y W y  e
W y 

– 

k k 1+ ek 1+
ke

k
– 

k 1=

n 1–

 e

e 1– 2
------------------- 1– e

n
1 n 2n

2
–+  ne

1 n+
1– n+  n

2
e

1– n+
+ + + .=


W y 



W y   u y kek– 
k 1=

1 1

--- WUi

y +

 = y 0 .

WUi
i 1 2    i

W y   1 10=

FIGURE 26.   Graph in the error in  for the case of .W y   0.1=

y

W y  W y –

 10
q–

=

10
q
W y 

W y  11q 1–+ q

W y  10= q 6=

10
7

75

https://doi.org/10.28924/ada/ma.2.14


 

  
Theorem 7.4.  Direct Evaluation of Lambert W with Specified Resolution. The Lambert W
function can be evaluated, with a maximum error of , according to

(109)

where  is defined in Theorem 7.2 and  is the  digit in the decimal
expansion of :

(110)

Proof. The floor of the Lambert W function has been defined in Theorem 7.2. Consider
the illustration shown in Figure 27. With

(111)

and with  being the  digit to the right of the decimal point, it follows that

(112)

Iteration with finer resolution, and from the point defined by , yields 

etc.        

 10
q–

=

W y  W y  d1 y   dq y + + + =

W y  10
q
dq y  qth

W y 

d1 y  1
10
------ u y W y  k

10
------+ W y  k

10
------+exp–

k 1=

10

= 

d2 y  1
100
--------- u y W y  d1 y  k

100
---------+ + W y  d1 y  k

100
---------+ +exp–

k 1=

10

 =



dq y  1

10
q

-------- u y W y  di y 
i 1=

q 1–

 k

10
q

--------+ + W y  di y 
i 1=

q 1–

 k

10
q

--------+ +exp–
k 1=

10

 .=

W y  W y  d1 y  d2 y  + + +=

dq y  qth

d1 y  1
10
------ u y W y  k

10
------+ W y  k

10
------+exp– .

k 1=

10

=

W y  d1 y + d2 y 

FIGURE 27.   Illustration of demarcation that underpins determination of , to

a set resolution of , between  and .

W y 

 ke
k

k 1+ ek 1+

y

k +

k 1+

k + ek +

k

ke
k

W y 

k 2+

k 2+ ek 2+

k 1+ ek 1+
k 1 –+ ek 1 –+

k 1 –+

https://doi.org/10.28924/ada/ma.2.14


 

  
The number of terms in the summation defined by Equation 109 comprises approxi-

mately  terms for the evaluation of , plus  terms for the summations
comprising  and  terms for the summation of , ,

.     

7.6.2.  Note. Theorem 7.4 defines a series for the Lambert W function, which, by
construction is convergent, i.e. 

(113)

and is such that

 (114)

8.  CONCLUSION

A geometric based approach for iteratively specifying approximations to the Lambert
W function, which can achieve any set relative error bound over the interval , was
detailed. The approximations are also valid for the interval  but are not sharp at
the point . Convergence was proved. For the interval , arbitrarily accurate
approximations, based on a two point spline approximation, were specified. Iteration,
either by using the iteration structure inherent in the definition of the Lambert W func-
tion, or via the Newton-Raphson method, leads to significantly improved approximations
albeit with increasing complex functional forms. 

Applications of the approximations were detailed and include, first, analytical expres-
sions for the Lambert W function that achieve set relative error bounds over the interval

. Second, based on the geometry inherent in the approximations, upper and lower
bounds for the Lambert W function that can be made arbitrary accurate. Third, higher
accuracy spline based approximations for the Lambert W function based on the defined
upper and lower bounded functions. Fourth, analytical expressions for the evaluation of

, and the integral of , without knowledge of  for . Finally, a
direct approach for evaluating the Lambert W function to achieve a prior defined error. 

Acknowledgement:   The author is pleased to acknowledge the support of Prof. A. Zoubir,
SPG, Technische Universität Darmstadt, Darmstadt, Germany, who hosted a visit where
the research, underpinning this paper, was completed.

APPENDIX A.  PROPERTIES OF LAMBERT W FUNCTION

The following are useful properties of the Lambert W function:

(115)

(116)

The latter formula underpins the iteration:

W y  W y  10q

d1 y  d2 y   dq y   q 1– d1 y  d2 y + 

d1 y  d2 y   dq y + + +

W y  W y  d1 y  d2 y  + + += y 0 

W y  W y  d1 y   dq y + + +– 10
q–
.

0  
1 e– 0 

1 e– 1 e– 0 

0 

W y  W y  W y  y 0  

W z z ln  z ln = z 0

W y  y
W y 
------------ln y ln W y  ln– = = y 0.

https://doi.org/10.28924/ada/ma.2.14


 

  (117)

To prove that , consider  which implies  and,
thus, .

The relationship  follows from the definitions , 
which implies  and, thus, .

A.1.  Differentiation. The derivatives of the Lambert W function are defined according to

(118)

(119)

where the second inequalities follow from the relationship  and the polyno-
mial  is defined according to

(120)

The coefficients in this expression are defined according to (https://oeis.org/A042977; [7],
eqn 3.4; [23], p. 1370):

(121)

Explicit expressions are:

(122)

APPENDIX B.  PROOF of Theorem 3.2

The iteration formula, as specified in Theorem 3.1, yields the first order approxima-
tions as stated in Equation 30:

(123)

W y  y ln y ln W y  ln– ln– =

W y  y ln y ln y ln W y  ln– ln– ln– =



W z z ln  z ln= f x  xe
x

= f z ln  z z ln=

z ln f
1–
z z ln  W z z ln = =

W y  y ln W y  ln–= y xe
x

= x W y =

y ln x x ln+= x y ln W y  ln–=

W
1 

y  e
W y –

1 W y +
---------------------- W y 

y 1 W y + 
------------------------------= =

W
k 

y 
W

k
y pk W y  

y
k

1 W y + 
2k 1–

--------------------------------------------
pk W y  e kW– y 

1 W y + 2k 1–
------------------------------------------= =

y W y eW y 
=

pk

pk r  ck 0 ck 1 r ck 2 r
2  ck k 1– r

k 1–
.+ + + +=

cn k

0 k 0

1– n 1+
n
n 1–

k 0=

n 1– cn 1– k 1–– 3 n 1–  k 1+ – cn 1– k– k 1+ cn 1– k 1++ 1 k n 3– 

n 1– cn 1– k 1–– 3 n 1–  k 1+ – cn 1– k– k n 2–=

n 1– cn 1– k 1–– n 1– != k n 1–=









=

p1 r  1= p2 r  2 r+ – = p3 r  9 8r 2r
2

+ + =

p4 r  64 79r 36r
2

6r
3

+ + + – = p5 r  625 974r 622r
2

192r
3

24r
4

+ + + + =

p6 r  7776 14543r 11758r
2

5126r
3

1200r
4

120r
5

+ + + + + .–=

WL1
y  y

1 y+
------------

n1 y 
d1 y 
-------------= =

WU1
y  1 y+ ln 1 ln– d1 y  ln

n1 y 
y

-------------ln– = =

https://doi.org/10.28924/ada/ma.2.14


 

  
where  and . The second order approximations, as specified by
Equation 31, can be written in the form 

(124)

where

(125)

The third order approximations, as specified by Equation 32 and Equation 33, can be
written in the form:

(126)

(127)

where

(128)

Thus, iteration yields the general formulas:

(129)

where

(130)

APPENDIX C.  PROOF of Theorem 3.5

Consider the geometry illustrated in Figure 15 and an initial approximation for ,

based on the intersection of the second order Taylor series for  at the origin, i.e.

n1 y  y= d1 y  1 y+=

WL2
y  y 1 1 y+ ln+ 

1 2y+
---------------------------------------

n2 y 
d2 y 
-------------= =

WU2
y  1 2y+ ln 1 1 y+ ln+ ln– d2 y  ln

n2 y 
y

-------------ln– = =

n2 y  y 1 1 y+ ln+  n1 y  1 d1 y  ln+ = =

d2 y  1 2y+ n1 y  d1 y .+= =

WL3
y 

y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+

1 3y y 1 y+ ln+ +
------------------------------------------------------------------------------------------------------------------------------------

n3 y 
d3 y 
-------------= =

WU3
y  1 3y y 1 y+ ln+ + ln 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+ln–=

d3 y  ln
n3 y 
y

-------------ln– =

n3 y  y 1 1 y+ ln+  1 1 2y+ ln 1 1 y+ ln+ ln–+=

n2 y  1 d2 y  ln
n2 y 
y

-------------ln–+ =

d3 y  1 3y y 1 y+ ln+ + n2 y  d2 y .+= =

WLi

ni y 
di y 
------------= WUi

di y  ln
ni y 
y

------------ln– =

ni y  ni 1– y  1 di 1– y  ln
ni 1– y 

y
-------------------ln–+ = n0 y  y= d0 y  1= 

di y  ni 1– y  di 1– y .+=

xo

ye
x–

https://doi.org/10.28924/ada/ma.2.14


 

  , and  which is . The problem with such an

approximation is that it only yields a real solution for . 
A practical approach is to utilize an affine approximation at the origin of  lead-

ing to the first approximation of . To establish a further approximation,

consider the point where the level  intersects  which is

(131)

A second level approximation follows by finding the intersection of a second order Taylor
series at the point , which is

(132)

with  to yield

(133)

and

(134)

Iteration in this manner leads to the general iteration formulas:

(135)

Simulation results (see Figure 16) indicate that the approximations also have good con-
vergence for the interval  but the approximations are not sharp at .

APPENDIX D.  PROOF of Theorem 4.1

Consider the case of  fixed, , and the illustration shown in Figure 18. By con-
struction, ,  and  for . Further, (see Equation 25)

(136)

and it follows that , , is a monotonically increasing sequence.

Using Equation 136 it follows that

y 1 x– x
2

2+  x WL1

1
y
--- 1 y 1 2y y

2
–++=

1 2– y 1 2+ 

y 1 x– 
WL1

y 1 y+ =

WL1
ye

x–

WU1

y
WL1

---------ln 1 y+ .ln= =

WU1

WL1
x WU1

– WL1
–

x WU1
– 2WL1

2
-------------------------------------+ 

x

WL2

1
WL1

--------- 1 WL1
1 WU1

+  1 WL1

2
– 2WL1

1 WU1
+ +–+=

WU2

y
WL2

--------- .ln=

WLi
1

WLi 1–

-------------- 1 WLi 1–
1 WUi 1–

+  1 WLi 1–

2
– 2WLi 1–

1 WUi 1–
+ +–+ =

WUi

y
WLi

---------ln .=

1 e– 0  y 1 e–=

y y 0
Li 0 Ui

0 WUi
WLi

 i 1 2   

WLi 1+
WLi

1 WUi
+

1 WLi
+

------------------- =

WLi
i 1 2   

https://doi.org/10.28924/ada/ma.2.14


 

  (137)

As , and , it follows that

(138)

where . Thus, as  monotonically increases with , it follows that  mono-

tonically decreases with , i.e. . It then follows that

(139)

Hence, convergence is guaranteed as . Thus:  and

. The result  implies that  and, thus,

. 

APPENDIX E.  ALTERNATIVE FORM FOR SPLINE APPROXIMATION

The general form for a  order, two point, spline approximation for a function , over
the interval , has been detailed in Howard, [16], eqn. 40. The assumption is that the
function  is at least  order differentiable over the interval . The approximation,
denoted , can be written in the modified form:

(140)

In this equation, the double summation can be rewritten by utilizing the transformations
 and , , . The possible values of  are

detailed in Table 8 and for  fixed, the valid values for  are from the set .    

TABLE 8.  Valid values of  for ,
.

i

k 0 1 2 3 ... n-2 n-1 n
0 0 1 2 3 n-2 n-1 n
1 1 2 3 4 n-1 n
2 2 3 4 5 n
3 3 4 5 6

Li Li 1+
– xo WLi

–  xo
WLi

1 WUi
+ 

1 WLi
+

---------------------------------––
WLi

WUi
WLi

– 

1 WLi
+

---------------------------------------.= =

WUi
WLi

– Li Ui
+= Ui

0

Li 1+
Li

WLi

1 WLi
+

------------------– Li Ui
+  1

WLi

1 WLi
+

------------------– Li riLi= =

ri
1

1 WLi
+

------------------= WLi
i ri

i 0 ri 1+ r
i

1 

0 Li 1+
L1

rk
k 1=

i

 L1
r1
i
.  

0 r1
1

1 y 1 y+ +
--------------------------------= 1  Lii 

lim 0=

WLii 
lim xo= Li 1+

Li

WLi

1 WLi
+

------------------– Li Ui
+ = Uii 

lim 0=

WUii 
lim xo=

nth f

  
f nth   
fn

fn x   x– n 1+

 – n 1+
---------------------------- f

k   
k!

---------------- n i+ !
i!n!

-----------------
i 0=

n k–

 x – k i+

 – i
--------------------------

k 0=

n

 +=

x – n 1+

 – n 1+
---------------------------- 1– kf k   

k!
------------------------------ n i+ !

i!n!
-----------------

i 0=

n k–

  x– k i+

 – i
------------------------- .

k 0=

n



r i k+= u i= k 0 1  n    i 0 1  n k–    r i k+=

r i 0 1  r   

r i k+= k 0 1  n   
i 0 1  n k–   

https://doi.org/10.28924/ada/ma.2.14


 

  

With ,  and , , Equation 140 can be written as

(141)

Thus:

(142)

where

(143)

APPENDIX F.  PROOF of Lemma 1

First, the definitions of ,  and  with , imply:

(144)

It then follows that  which implies . With

 and , the required result of  then follows.
Second, the transformation of , , results in

(145)

As  it then follows that  and the final result follows:

(146)

n-2 n-2 n-1 n
n-1 n-1 n
n n

TABLE 8.  Valid values of  for ,
.

i

k 0 1 2 3 ... n-2 n-1 n

r i k+= k 0 1  n   
i 0 1  n k–   



r 0 1  n    u 0 1  r    i u= k r u–=

fn x   x– n 1+

 – n 1+
---------------------------- x – r f

r u–   
r u– !

----------------------- n u+ !
u!n!

-------------------
u 0=

r

 1

 – u
--------------------

r 0=

n

 +=

x – n 1+

 – n 1+
----------------------------  x– r 1– r u–

f
r u–   

r u– !
-------------------------------------------- n u+ !

u!n!
-------------------

u 0=

r

 1

 – u
-------------------- .

r 0=

n



fn x   x– n 1+
an r x – r

r 0=

n

 x – n 1+
bn r  x– r

r 0=

n

+=

an r
1

 – n 1+
---------------------------- f

r u–   
r u– !

----------------------- n u+ !
u!n!

-------------------
u 0=

r

 1

 – u
-------------------- =

bn r
1

 – n 1+
---------------------------- 1– r u–

f
r u–   

r u– !
-------------------------------------------- n u+ !

u!n!
-------------------

u 0=

r

 1

 – u
--------------------. =

f g1 x x1 1–= x 1–

y1 g1 x1  1
e
--- x1 1– e

x1 1–
+ = x1 0.=

y1 1 e– f x1 1– = f
1–
y1 1 e–  x1 1–=

x1 g1
1–
y1 = y y1 1 e–= f

1–
y  g1

1–
y 1 e+  1–=

y2 g x1  g1 x1  y1= = = x1 0

g
1–
y2  x1 g

1–
y1 .= =

x1 g1
1–
y1 = g1

1–
y1  g

1–
y1 =

f
1–
y  g1

1–
y 1

e
---+ 1– g

1–
y 1

e
---+ 1.–= =

https://doi.org/10.28924/ada/ma.2.14


 

  APPENDIX G.  PROOF of Theorem 5.1

With  denoting the differentiation operator, the following well known results apply
for an arbitrary function :

(147)

(148)

(149)

etc. 
Consider , as defined by Equation 75, over the interval . A spline approxima-

tion (see Equation 70) for , of order , requires derivatives, of orders zero to  at the
points  and , to be determined. Using the above formulas, such values can be
determined from the derivatives of  at the points  and  and values of these deriva-
tives are tabulated in Table 9. To determine the derivative values at zero, the standard
Taylor series expansion for the exponential function can be used to yield the alternative
form for  of

(150)

Using Equation 70, and the derivative values given in Table 9, the spline approximations
for , based on the points  and  and for orders one to four, are:

(151)

(152)

D

f

D f
1–
z   1

f
1 

x 
----------------

x f
1–
z =

= D
2 

f
1–
z   f

2 
x –

f
1 

x  
3

-----------------------

x f
1–
z =

=

D
3 

f
1–
z   f

3 
x –

f
1 

x  
4

-----------------------
3 f

2 
x  

2

f
1 

x  
5

--------------------------+

x f
1–
z =

=

D
4 

f
1–
z   f

4 
x –

f
1 

x  
5

----------------------- 10f
3 

x f 2 
x 

f
1 

x  
6

--------------------------------------
15 f

2 
x  

3

f
1 

x  
7

-----------------------------–+

x f
1–
z =

=

g 0 1 

g
1–

n n

0 1 e
g 0 1

g

g x1 
x1

2e
---------- 1 2

i 1+ x1
i

i 2+ !
---------------------

i 1=



+ .=

g
1–
y2  0 1 e

g1
1–
y2  2ey2 1 2 ey2 1 1

2e
---------- 3

2 2
----------–+– ey2

2
1 2– 2

e
-------++=

g2
1–
y2  2ey2

1
2ey2

3
---------------– 6e 2 1–  7

2
-------– 2 2

e
----------– y2

2
–+

e
3 2 17

2
------- 8– 6 2e– 4 2

e
----------– y2

3
e

2 10 2
3

------------- 3– 5e

2
-------– 2 2– y2

4
+

=

https://doi.org/10.28924/ada/ma.2.14


 

  

  

(153)

(154)

In general:

(155)

TABLE 9.  Values of the derivatives of  at the points zero and one. 

order

0 0

1

2

3

4

5

g x1 

g
i 

0  g
i 

1 

1

e
------

1

2e
---------- e

2
------

2

3 e
---------- e 1 e

4
---–

5

12 2e
---------------- 3 e

2
---------- 1 e– e

2

4
-----+

11

45 2e
---------------- 2 e 1 3e– 9e

2

4
-------- 15e

3

32
-----------–+

59

432 2e
------------------- 5 e

2
---------- 1 8e– 27e

2

2
----------- 15e

3

2
-----------– 21e

4

16
-----------+ +

g3
1–
y2  2ey2

1
2ey2

3
---------------–

11ey2
2

36
-------------- e

3 2 191
9

--------- 125

3 2
----------– 25 e

2
------------- 8 2

e
---------- 6 2

e
3 2

----------+ + + y2
3

+–+

e
2 281

6
--------- 146 2

3
----------------– 32 2e 22 2 18 2

e
-------------+ + + y2

4
–

e
5 2 335

9
--------- 40 2– 55e

3 2

2
----------------- 20 2 e 18 2

e
-------------+ + + y2

5
+

e
3 371

36
--------- 34 2

3
-------------– 8 2e

2
6 2e 6 2+ + + y2

6

=

g4
1–
y2  2ey2

1
2ey2

3
---------------–

11ey2
2

36
--------------

43e
3 2

y2
3

135 2
----------------------–+ +

e
2 4075

27 2
------------- 895

12
---------– 91e

2
---------– 61

2
-------– 27 2

e
-------------– 64 2

3e
2

-------------– y2
4

–

e
5 2 6658 2

27
------------------- 2126

9
------------– 315e

3 2

2
--------------------– 110 2e– 102 2

e
----------------– 256 2

3e
3 2

----------------– y2
5

+

e
3 8467 2

27
------------------- 1175

4
------------– 207 2e

2
– 149 2e– 144 2– 128 2

e
----------------– y2

6
–

e
7 2 4904 2

27
------------------- 502

3
---------– 245e

5 2

2
--------------------– 90 2e

3 2
– 90 2e– 256 2

3 e
----------------– y2

7
+

e
4 10843

135 2
---------------- 1315

36
------------– 55e

3

2
-----------– 41e

2

2
-----------– 21 2e– 64 2

3
-------------– y2

8

=

gk
1–
y2  2ey2 1 1y2 2y2

2 3y2
3 4y2

4  2ky2
2k

+ + + + + + 

https://doi.org/10.28924/ada/ma.2.14


by Equation 70 with , based on the points  and

 

  
for appropriately defined constants and, thus:

(156)

APPENDIX H.   PROOF of Theorem 7.1

A zero order spline approximation is simply an affine approximation between the two
specified points. Consistent with Figure 23, the zero order spline approximation to ,
denoted , is an affine approximation between the points  and

. Thus:

(157)

With the approximation  it follows that

(158)

Simplification yields

(159)

Substitution of  and  yields the required result.

H.1.  General Result. The general result arises from the spline approximation, specified

 where , , and with

 (160)

Here  is defined by Equation 120. The  order spline approximation, for
, is

W y  g
1–

y 1
e
---+ 1–=

2e y 1
e
---+ 1 1 y 1

e
---+ 2 y 1

e
---+ 3 y 1

e
---+

3 2
 2k y 1

e
---+

k
+ + + + + 1.–

W y 
f0 uo uo exp uo 

vo vo exp vo 

f0 y  uo y uo uo exp– 
vo uo–

vo vo exp uo uo exp–
--------------------------------------------------------- y uo uo exp vo vo exp .+=

xo W yo = f0 yo 

xo uo yo uo uo exp– 
vo uo–

vo vo exp uo uo exp–
---------------------------------------------------------.+

xo
uovo vo exp uo exp–  yo vo uo– +

vo vo exp uo uo exp–
------------------------------------------------------------------------------------------------.

uo WLi
yo = vo WUi

yo =

f y  W y = uo uo exp uo 

vo vo exp vo  uo WLi
yo = vo WUi

yo =

W
1 

y  e
W y –

1 W y +
----------------------= W

k 
y 

pk W y  e kW– y 

1 W y + 2k 1–
------------------------------------------= k 1 2   .

pk nth

y uo uo exp vo vo exp 

https://doi.org/10.28924/ada/ma.2.14


 

  

(161)

The results ,  imply that

 (162)

assuming . Hence:

(163)

The required result follows: the approximation for , denoted , arises for the
case of . Thus, . 

fn y 
vo vo exp y– n 1+

vo vo exp uo uo exp– n 1+
------------------------------------------------------------------------ =

y uo uo exp– r
W

r u– 
uo uo exp 

r u– !
------------------------------------------------- n u+ !

u!n!
-------------------

u 0=

r

 1

vo vo exp uo uo exp– u
-----------------------------------------------------------------

r 0=

n

 +

y uo uo exp– n 1+

vo vo exp uo uo exp– n 1+
------------------------------------------------------------------------ 

vo vo exp y– r
1– r u–

W
r u– 

vo vo exp 
r u– !

--------------------------------------------------------------------- n u+ !
u!n!

-------------------
u 0=

r

 1

vo vo exp uo uo exp– u
-----------------------------------------------------------------

r 0=

n



W uo uo exp  uo= W vo vo exp  vo=

W
r u– 

uo uo exp 
pr u– uo e

r u– uo–

1 uo+ 2 r u–  1–
----------------------------------------------=

W
r u– 

vo vo exp 
pr u– vo e

r u– vo–

1 vo+ 2 r u–  1–
----------------------------------------------=

r u

fn y 
vo vo exp y– n 1+

vo vo exp uo uo exp– n 1+
------------------------------------------------------------------------ =

y uo uo exp– r

n r+ !uo

r!n! vo vo exp uo uo exp– r
-------------------------------------------------------------------------- +

pr u– uo e
r u– uo–

r u– ! 1 uo+ 2 r u–  1–
------------------------------------------------------------- n u+ !

u!n!
-------------------

u 0=

r 1–

 1

vo vo exp uo uo exp– u
-----------------------------------------------------------------


r 0=

n

 +

y uo uo exp– n 1+

vo vo exp uo uo exp– n 1+
------------------------------------------------------------------------ 

vo vo exp y– r

n r+ !vo

r!n! vo vo exp uo uo exp– r
-------------------------------------------------------------------------- +

1– r u–
pr u– vo e

r u– vo–

r u– ! 1 vo+ 2 r u–  1–
------------------------------------------------------------------ n u+ !

u!n!
-------------------

u 0=

r 1–

 1

vo vo exp uo uo exp– u
-----------------------------------------------------------------


r 0=

n



W yo  Wn i yo 

y yo= Wn i yo  fn yo =

https://doi.org/10.28924/ada/ma.2.14


 

 

 

 

 

 

 

 

 

 

 

 

 

 

https://doi.org/10.28924/ada/ma.2.14
https://doi.org/10.1080/10652469.2018.1528247
https://doi.org/10.1109/81.895330
https://doi.org/10.1016/S0378-4754(00)00172-5
https://doi.org/10.1007/s10910-018-0932-3
https://doi.org/10.1007/s10910-018-0932-3
https://doi.org/10.1016/S0893-9659(98)00097-4
https://doi.org/10.1007/BF02124750
https://doi.org/10.1155/2021/6695559
https://doi.org/10.1017/S0022377805003788
https://doi.org/10.30538/oms2021.0149
https://doi.org/10.1016/j.cam.2012.11.021
https://scholar.google.com/citations?user=invpkckAAAAJ&hl=en&oi=sra


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

https://doi.org/10.28924/ada/ma.2.14
https://doi.org/10.1016/j.bej.2012.01.010
https://doi.org/10.3390/mca24020035
https://doi.org/10.1007/s10444-017-9530-3
https://doi.org/10.1016/j.physleta.2015.12.004
https://doi.org/10.1111/2041-210X.12568
https://doi.org/10.1088/0143-0807/36/3/035030
http://arxiv.org/abs/1408.3999
https://doi.org/10.1093/imamat/hxu057
https://doi.org/10.1139/p00-065
https://doi.org/10.1016/j.cpc.2012.07.008
https://doi.org/10.3390/math6040056



